Abstract
Under suitable conditions, a function f(t) in a principal shift-invariant space can be recovered from its uniform samples { f(n)}
n∈ℤ with a simple sampling formula. Provided that the generator ϕ has compact support, we consider the sampling problem in the bigger space V
ϕ ≔ {∑a
n
ϕ(t − n) : a
n
∈ ℂℤ}. In this space, there exist infinite functions with the same samples y
n
= f(n). We show that polynomial growth conditions give the uniqueness: if y
n
has polynomial growth, there is a unique function of polynomial growth f ∈ V
ϕ, satisfying f(n) = y
n
, n ∈ ℤ. This function is given by the known sampling formula. The same result is proved also when we consider average samples y
n
= f ∗ h(n).
Acknowledgements
This work has been supported by the grant MTM2009–08345 from the D.G.I. of the Spanish Ministerio de Ciencia y Tecnología.